KYUNGPOOK Math. J. 2019; 59(2): 325-333
Published online June 23, 2019
Copyright © Kyungpook Mathematical Journal.
Some Generating Relations of Extended Mittag-Leﬄer Functions
Nabiullah Khan, Mohd Ghayasuddin, Mohd Shadab∗
Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India
e-mail : firstname.lastname@example.org
Department of Mathematics, Faculty of Science, Integral University, Lucknow226026, India
e-mail : email@example.com
Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia(A Central University), New Delhi-110025, India
e-mail : firstname.lastname@example.org
Received: March 16, 2017; Revised: March 12, 2019; Accepted: March 18, 2019
Motivated by the results on generating functions investigated by H. Exton and many other authors, we derive certain (presumably) new generating functions for generalized Mittag-Leffler-type functions. Specifically, we introduce a new class of generating relations (which are partly bilateral and partly unilateral) involving the generalized Mittag-Leffler function. Also we present some special cases of our main result.
Keywords: generalized Mittag-Leﬄer’s function, hypergeometric function, generating function.
In 1903, the Swedish mathematician Gosta Mittag-Leffler  introduced the function
The Mittag-Leffler function is a direct generalization of the exponential function to which it reduces when
Wiman  introduced a new generalization of
which is known as the Wiman function. Properties of the Wiman function
Prabhakar  introduced a further generalization of
In a sequel to the above-mentioned works, Shukla and Prajapati  defined the following generalization of the Mittag-Leffler function:
Subsequently, Khan and Ahmad  defined the following two interesting generalizations of these functions and investigated their associated properties:
2. Generating Relation
On expanding the function
in series form, we obtain
which is our required result.
3. Special Cases
(1) On setting
(2) On setting
(3) On setting
(5) On setting
(6) On setting
(7) On setting
(8) On setting
(9) On setting
(10) On setting
(11) On setting
4. Concluding Remark
In our present investigation, we have studied a number of generating functions for the extended Mittag-Leffler-type functions given in [4, 6, 13, 14, 15]. The main generating function is the further generalization of the result given by Kamarujjama and Khan . The results of this paper, especially (
The authors thank the constructive comments and suggestions by anonymous referees. They have contributed to improve the presentation of this manuscript. The authors wish to acknowledge R. B. Paris (Abertay University, Dundee, UK) for his assistance with the improvement of the text.
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